What is the limit as x approaches 0 of 2(1+3x)^{2/x} ?

The limit as x approaches 0 of 2(1+3x)^{2/x} is 2e^6

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limit as x approaches 0 of 2(1+3x)^{2/x}

x \to 0 lim (2 (1 + 3 x)^{\frac{2}{x}})

2 e^{6}

Decimal

806.85758 \dots

Solution steps

x \to 0 lim (2 (1 + 3 x)^{\frac{2}{x}})

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

= 2 \cdot x \to 0 lim ((1 + 3 x)^{\frac{2}{x}})

Apply exponent rules

= 2 \cdot x \to 0 lim (e^{\frac{2}{x} l n (1 + 3 x)})

Apply the Limit Chain Rule:

e^{6}

= 2 e^{6}

What is the limit as x approaches 0 of 2(1+3x)^{2/x} ?
The limit as x approaches 0 of 2(1+3x)^{2/x} is 2e^6